Everythingers might just be interested in an extract from an email forming part of a recent brief discussion with John Leslie, who in his book 'Universes' (sect 4.69), dismisses the idea of all logically possible universes because it seems to entail the failure of induction (ie failure of some known physical laws).
This is the extract:
A defence against the induction failure challenge to the all logically
possible universes hypothesis (ALPUH):
1. The aim is to obtain a general idea of how numbers of indistinguishable
universes vary under any reasonable ALPUH. This general idea only needs a
level of precision and plausibility that is sufficient to cast serious doubt
on the induction failure challenge to that ALPUH. [Names used elsewhere for
this challenge: white rabbit problem, dragon universes problem.] (Note that
there can in principle be different ALPUH's according to how many copies of
indistinguishable universes that they predict.)
2. We only need to consider the failure or otherwise of induction in those
universes which contain thinking entities of our general kind ('self-aware
structures' in Tegmark's terminology).
3. There are at least two general categories of ways to represent our
universe. The first category is directly related to how we perceive it: this
can be in terms, say, of percepts, or of objects embedded in space-time. (It
is this way that is at the basis of the induction failure challenge.) The
second category entails directly incorporating the physical laws that govern
some or all of our universe. An example of such a way is a mathematical
model of our universe (incorporating initial or boundary conditions). Such a
formal system would be expected to be ultimately grounded in axioms, from
which the theorems (specifying everything in the universe governed by the
laws) would be derivable by simple rules of inference. Any events/entities
in the universe not entirely governed by the laws (or, if one is a strict
physicalist: any events/entities in a similar kind of universe that are not
entirely governed by physical laws) would have to be incorporated in the
mathematical representation by additional axioms (perhaps comprising in some
way a simple list of its properties), and/or by adding new rules of
inference, and/or (most likely) including adjustments to the existing
axioms/inference rules. Note however, that all of these increase the
complexity of the representation.
4. In any naturally occuring all logical possible universes scenario, one
would expect the distribution of unverses to be more likely to be closer to,
or actually reflect, the second category of representations than the first,
partly because the second explicitly makes allowance for the physical laws,
and partly because the direct view of human beings of the world (the first
category) is governed by how we happen to have evolved to perceive it, which
in turn is governed by the requirements of how to survive in our
environment, and what features of the environment biological creatures are
physically able to perceive. It is likely to be 'anthropically biassed'.
5. We will pursue the idea of 'all possible axiomatic formal systems' as an
example of the second category, but bearing in mind that this may only be
one possibility for this category.
6. Formal systems are expressible in terms of fundamental units - usually
the symbol (we can add a delimiter symbol to separate axioms). So a symbol
string of axioms, plus standard rules of inference, if well-formed and
providing consistent theorems, will comprise a theory, which may specify one
(or more than one) universe. We can have as many axioms (and inference
rules - though it is simplest to keep these fixed if possible) as we like.
Now all possible universes that are expressible by any formal system must be
found within all possible combinations of symbols (or all possible axioms -
this is easier to imagine).
7. Let us suppose that our universe (A) is minimally specifiable by m
axioms; and another universe (B), which is identical to ours except that
induction apparently fails at some future moment by the physical laws
suddenly changing, is minimally specifiable by m+p axioms. (Strictly
speaking this is misleading because we can't just specify a change in a law
by adding some axioms, but the point is that B would need more axioms than
for A.)
8. Now if we consider all possible combinations of axioms up to n, where n
is any number greater than m+p, we find that universes indistinguishable
from A must greatly outnumber those indistinguishable from B by the
following reasoning. Between m+p and n we have many combinations of
axioms available - these will not (for the most part) be able to include
variables already functionally involved in A or B (because as theories they
are already consistent), but they can include axioms involving entirely new
variables - effectively specifying a different universe, or entity. For this
range (m+p to n), the effect is to multiply up the number of examples of A
and B by roughly the same amount - so this effect more or less cancels
through. However for the range m to m+p we have a multiplier for A, but not
for B (because B already requires axioms in this range to be of a certain
type). So in total there are more examples of A than of B.
9. Bearing in mind [similar] results from other approaches falling within the second
category, it seems plausible to suppose that for whatever logical unit
(axioms/bits/symbols etc) is most applicable to the actual provision of all
logically possible universes, the same underlying method will still hold
good: simpler universes will predominate. The result explains why Ockham's
Razor is applicable to our physical world and also why we don't find
unambiguous cases of paranormal events: our world is the simplest possible
consistent with the existence of self-aware structures, if the hypothesis is
correct. Induction will not fail in general, because such failures would
entail a more complex universe, which is less frequently represented (and so
we are unlikely to be in it).
Alastair Malcolm
(More details: see web pages starting at
http://www.physica.freeserve.co.uk/p105.htm)
Received on Fri May 12 2000 - 04:54:16 PDT